Idea Transcript
Remarks concerning the status & some ramifications of
EHRENFEST’S THEOREM Nicholas Wheeler, Reed College Physics Department March 1998
Introduction & motivation. Folklore alleges, and in some texts it is explicitly—
if, as will emerge, not quite correctly—asserted, that “quantum mechanical expectation values obey Newton’s second law.” The pretty point here at issue was first remarked by Paul Ehrenfest (–), in a paper scarcely more than two pages long.1 Concerning the substance and impact of that little gem, Max Jammer, at p. 363 in his The Conceptual Development of Quantum Mechanics (), has this to say: “That for the harmonic oscillator wave mechanics agrees with ordinary mechanics had already been shown by Schr¨ odinger. . . 2 A more general and direct line of connection between quantum mechanics and Newtonian mechanics was established in 1927 by Ehrenfest, who showed ‘by a short elementary calculation without approximations’ that the expectation value of the time derivative of the momentum is equal to the expectation value of the negative gradient of the potential energy function. Ehrenfest’s affirmation of Newton’s second law in the sense of averages taken over the wave packet had a great appeal to many physicists and did much to further the acceptance of the theory. For it made it possible to describe the particle by a localized wave packet which, though eventually spreading out in space, follows the trajectory of the classical motion. As emphasized in a different context elsewhere,3 Ehrenfest’s theorem 1
“Bemerkung u ¨ber die angen¨ aherte G¨ ultigkeit der klassichen Machanik innerhalb der Quanatenmechanik,” Z. Physik 45, 455–457 (1927). 2 ¨ Jammer alludes at this point to Schr¨ odinger’s “Der stetige Ubergang von der Mikro- zur Makromechanik,” Die Naturwissenschaften 28, 664 (1926), which in English translation (under the title “The continuous transition from micro- to macro-mechanics”) appears as Chapter 3 in the 3rd (augmented) English edition of Schr¨ odinger’s Collected Papers on Wave Mechanics (). 3 See Jammer’s Concepts of Mass (), p. 207.
2
Status of Ehrenfest’s Theorem
and its generalizations by Ruark 4 . . . do not conceptually reduce quantum dynamics to Newtonian physics. They merely establish an analogy—though a remarkable one in view of the fact that, owing to the absence of a superposition principle in classical mechanics, quantum mechanics and classical dynamics are built on fundamentally different foundations.” “Ehrenfest’s theorem” is indexed in most quantum texts,5 though the celebrated authors of some classic monographs6 have (so far as I have been able to determine, and for reasons not clear to me) elected pass over the subject in silence. The authors of the texts just cited have been content simply to rehearse Ehrenfest’s original argument, and to phrase their interpretive remarks so casually as to risk (or in several cases to invite) misunderstanding. Of more lively interest to me at present are the mathematically/interpretively more searching discussions which can be found in Chapter 6 of A. Messiah’s Quantum Mechanics () and Chapter 15 of L. E. Ballentine’s Quantum Mechanics (). Also of interest will be the curious argument introduced by David Bohm in §9.26 of his Quantum Theory (): there Bohm uses Ehrenfest’s theorem “backwards” to infer the necessary structure of the Schr¨ odinger equation. I am motivated to reexamine Ehrenfest’s accomplishment by my hope (not yet ripe enough to be called an expectation) that it may serve to illuminate the puzzle which I may phrase this way: I look about me, in this allegedly “quantum mechanical world,” and see objects moving classically along well-defined trajectories. How does this come to be so? I have incidental interest also some mathematical ramifications of Ehrenfest’s theorem in connection with which I am unable to cite references in the published literature. Some of those come instantly into focus when one looks to the general context within which Ehrenfest’s argument is situated. 4
The allusion here is to A. E. Ruark, “. . . the force equation and the virial theorem in wave mechanics,” Phys. Rev. 31, 533 (1928). 5 See E. C. Kemble, The Fundamental Principles of Quantum Mechanics (), p. 49; L. I. Schiff, Quantum Mechanics (3rd edition, ), p. 28; E. Mertzbacher, Quantum Mechanics (2nd edition, ), p. 41; J. L. Powell & B. Crassmann, Quantum Mechanics (), p. 98; D. J. Griffiths, Introduction to Quantum Mechanics (), pp. 17, 43, 71, 150, 162 & 175. Of the authors cited, only Griffiths draws recurrent attention to concrete applications of Ehrenfest’s theorem. 6 I have here in mind P. A. M. Dirac’s The Principles of Quantum Mechanics (revised 4th edition, ) and L. D. Landau & E. M. Lifshitz’ Quantum Mechanics (). W. Pauli’s Wellenmechanik () is a reprint of his famous Handbuch article, which appeared—incredibly—in , which is to say: too early to contain any reference to Ehrenfest’s accomplishment.
3
General observations concerning the motion of moments
1. Quantum motion of moments: general principles. Let |ψ) signify the state of a
quantum system with Hamiltonian H, and let A refer to some time-independent observable.7 The expected mean of a series of A-measurements can, by standard quantum theory, be described A = (ψ|A|ψ) and the time-derivative of A—whether one works in the Schr¨ odinger picture,8 the Heisenberg picture,9 or any intermediate picture—is given therefore by d dt A
1 i AH
=
− HA
(1)
Ehrenfest himself looked to one-dimensional systems of type H≡
1 2 2m p
−V
with V ≡ V (x)
and confined himself to a single instance of (1): d dt p
= =
1 i pH 1 i pV
− Hp − Vp
Familiarly [x, p ] = i 1
=⇒
[xn , p ] = i · n xn−1
whence
[ V (x), p ] = i · V (x)
so with Ehrenfest we have
d dt p
= −V (x)
d dt x
=
(2.1)
A similar argument supplies 1 m p
(2.2)
though Ehrenfest did not draw explicit attention to this fact. Equation (2.1) is notationally reminiscent of Newton’s 2nd law
p˙ = −V (x) with p ≡ mx˙ and equations (2) are jointly reminiscent of the first-order “canonical equations of motion” 1 x˙ = m p (3) p˙ = −V (x) 7
I will be using sans serif boldface type to distinguish operators (q-numbers) from real/complex numbers (c-numbers). d 1 8 A is constant, but |ψ) moves: dt |ψ) = i H|ψ). 9
|ψ) is constant, but A moves:
d dt A
=
1 i [A, H].
4
Status of Ehrenfest’s Theorem
1 2 that associate classically with systems of type H(x, p) = 2m p + V (x). But except under special circumstances which favor the replacement
V (x)
−→
V (x)
(3)
the systems (2) and (3) pose profoundly different mathematical and interpretive problems. Whence Jammer’s careful use of the word “analogy,” and of the careful writing (and, in its absence, of the risk of confusion) in some of the texts to which I have referred. The simplest way to achieve (3) comes into view when one looks to the case of a harmonic oscillator. Then V (x) = mω 2 x is linear in x, (3) reduces to a triviality, and from (2) one obtains d dt p d dt x
= −mω 2 x =
(4)
1 m p
For harmonic oscillators it is true in every case (i.e., without the imposition of restrictions upon |ψ)) that the expectation values x and p move classically. The failure of (3) can, in the general case (i.e., when V (x) is not quadratic), be attributed to the circumstance that for most distributions xn = xn . It becomes in this light natural to ask: What conditions on the distribution function P (x) ≡ ψ ∗ (x)ψ(x) are necessary and sufficient to insure that xn and xn are (for all n) equal? Introducing the so-called “characteristic function” (or “moment generating function”) Φ(k) ≡
∞ 1 (ik)n xn = eikx P (x)dx n! n=0
we observe that if xn = xn (all n) then Φ(k) = eikx , and therefore that P (x) =
1 2π
e−ik[x−x] dk = δ(x − x)
It was this elementary fact which led Ehrenfest to his central point, which (assuming V (x) to be now arbitrary) can be phrased as follows: If and to the extent that P (x) is δ-function-like (refers, that is to say, to a narrowly confined wave packet), to that extent the exact equations (2) can be approximated d dt p d dt x
= −V (x) =
1 m p
(5)
and in that approximation the means x and p move classically. But while P (x) = δ(x − x) may hold initially (as it is often assumed to do), such an equation cannot, according to orthodox quantum mechanics,
5
Free particle
persist, for functions of the form satisfy the Schr¨ odinger equation.
δ(x − xclassical (t))eiα(x,t) cannot be made to
2. Example: the free particle. To gain insight into the rate at which P (x) loses
its youthfully slender figure—the rate, that is to say, at which the equations xn = xn lose their presumed initial validity—one looks naturally to the time-derivatives of the “centered moments” (x − x)n , and more particularly to the leading (and most tractable) case n = 2. From (x − x)2 = x2 − x2 it follows that 2 2 d d d (6) dt (x − x) = dt x − 2x dt x To illustrate the pattern of the implied calculation we look initially to the case 1 of a free particle: H = 2m p2 . From (2) we learn that d dt p d dt x
= 0 so p is a constant; call it p ≡ mv =v ⇓
x = x0 + vt where x0 ≡ xinitial is a constant of integration
(7)
Looking now to the leading term on the right side of (6), we by (1) have 2 d dt x
=
2 2 1 2im [x , p ]
The fundamental commutation rule [AB, C] = A[B, C] + [A, C]B implies (and can be recovered as a special consequence of) the identity [AB, CD] = AC[B, D] + A[B, C]D + C[A, D]B + [A, C]DB with the aid of which we obtain [x2 , p2 ] = 2i(xp + px), giving 2 d dt x
=
1 m (xp
+ px)
(8)
Shifting our attention momentarily from x2 to (xp + px), in which we have now an acquired interest, we by an identical argument have d dt (xp
+ px) = =
1 2im [(xp 2 2 m p
+ px), p2 ] (9)
and are led to divert our attention once again, from (xp + px) to p2 . But 2 d dt p
=
2 2 1 2im [p , p ] 2
= 0 so p is a constant; call it m2 u2 by an argument that serves in fact to establish that For a free particle pn is constant for all values of n.
(10)
6
Status of Ehrenfest’s Theorem
Returning with this information to (9) we obtain (xp + px) = 2mu2 t+a a ≡ (xp + px)initial is a constant of integration which when introduced into (8) gives x2 =
1 m
mu2 t2 + at +s2 s2 ≡ x2 initial is a final constant of integration
We conclude that the time-dependence of the centered 2nd moments of a free particle can be described σp2 (t) ≡ (p − p)2 = m2 (u2 − v 2 ) 2 2 1 σx2 (t) ≡ (x − x)2 = m mu t + at + s2 − (x0 + vt)2
(11.1)
= (u2 − v 2 )t2 +
(11.2)
1 m (a
− 2mvx0 )t + (s2 − x20 )
Concerning the constants which enter into the formulation of these results, we note that x0 and s have the dimensionality of length u and v have the dimensionality of velocity a has the dimensionality of action and that the values assignable to those constants are subject to some constraint: necessarily (whether one argues from σp2 ≥ 0 or from σx2 (t → ±∞) ≥ 0) u2 − v 2 ≥ 0 while σx2 (0) ≥ 0 entails
s2 − x20 ≥ 0
A graph of σx2 (t) has the form of an up-turned parabola or is linear according as u2 − v 2 ≥ 0; the latter circumstance is admissible only if a − 2mvx0 = 0, but in the former case the requirement that the roots of σx2 (t) = 0 be not real and distinct (i.e., that they be either coincident or imaginary) leads to a sharpened refinement of that admissibility condition: (u2 − v 2 )(s2 − x20 ) −
1 2m (a
2 − 2mvx0 ) ≥ 0
(12)
By quick calculation we find (proceeding from (11.2)) that the least value ever assumed by σx2 (t) can be described
σx2 (t)
least
1 2 (a − 2mvx0 ) (u2 − v 2 )(s2 − x20 ) − 2m = (u2 − v 2 )
Free particle
7
and so obtain
1 σp2 (t)σx2 (t) = m2 (u2 − v 2 ) (u2 − v 2 )t2 + m (a − 2mvx0 )t + (s2 − x20 ) 2 ≥ m2 (u2 − v 2 )(s2 − x20 ) − 12 (a − 2mvx0 ) (13)
In deriving (13) we drew upon the principles of quantum dynamics, as 1 they refer to the system H = 2m p2 , but imposed no restrictive assumption upon the properties of |ψ); in particular, we did not (as Ehrenfest himself did) assume (x|ψ) ≡ ψ(x) to be Gaussian. A rather different result was achieved by Schr¨ odinger by an argument which draws not at all upon dynamics (it exploits little more than the definition A ≡ (ψ|A|ψ) and Schwarz’ inequality); if A and B refer to arbitrary observables, and |ψ) to an arbitrary state, then according to Schr¨ odinger10 AB − BA 2 AB + BA 2 (∆A)2 (∆B)2 ≥ + (14) − AB 2i 2 AB − BA 2 ≥ 2i which in a particular case (A → x, B → p) entails
xp + px 2 σp2 (t)σx2 (t) ≥ (/2)2 + − xp 2 Reverting to our established notation, we find
2 2 etc. = m(u2 − v 2 )t + 12 (a − 2mvx0 )
(15)
and observe that the expression on the right invariably vanishes once, at time a − 2mvx 0 t=− 2m(u2 − v 2 ) Which is precisely the time at which, according to (11.2), σx2 (t) assumes its least value. Evidently (13) will be consistent with (15) if and only if we impose upon the parameters {x0 , s, u, v and a} this sharpened—and non-dynamically motivated—refinement of (12): 1 2 (u2 − v 2 )(s2 − x20 ) − 2m (a − 2mvx0 ) ≥ (/2m)2 (16) Notice that we recover (12) if we approach the limit that m → ∞ in such a way 1 as to preserve the finitude of 2m (a − 2mvx0 ). 10
“Zum Heisenbergschen Unscharfenprinzip,” Berliner Berichte, 296 (1930). For discussion which serves to place Schr¨ odinger’s result in context, see §7.1 in Jammer’s The Conceptual Development of Quantum Mechanics (). For a more technical discussion which emphasizes the importance of the “correlation term” {etc.}—a term which the argument which appears on p. 109 of Griffiths’ text appears to have been designed to circumvent—see the early sections in Bohm’s Chapter 10. Or see my own quantum mechanics (), Chapter III, pp. 51–58.
8
Status of Ehrenfest’s Theorem
3. A still simpler example: the “photon”. We are in the habit of thinking of the
free particle as the “simplest possible” dynamical system. But at present we are concerned with certain algebraic aspects of quantum dynamics, and from that point of view it becomes natural to consider the Hamiltonian H = cp
(17)
which depends not quadratically but only linearly upon p. We understand c to be a constant with the dimensionality of velocity.11 Classically, the canonical equations of motion read x˙ = c and p˙ = 0 (18) and give x(t) = x0 + ct and p(t) = constant The entities to which the theory refers (lacking any grounds on which to call them “particles,” I will call them “photons”) invariably move to the right with speed c. Quantum mechanically, Ehrenfest’s theorem gives d dt p
=0
and
d dt x
=
1 i [x, cp ]
=c
which exactly reproduce the classical equations (17), and inform us that the 1st moments x and p move “classically:” x = x0 + ct p = constant: call it p Looking to the higher moments, the argument which gave (10) now supplies the information that that indeed pn is constant for all values of n, and so also therefore are all the centered moments of momentum; so also, in particular, is σp2 (t) = constant; call it P 2 From 2 d dt x
=
2 1 i [x , cp ]
= 2cx = 2c(x0 + ct)
we obtain x2 = s2 + 2cx0 t + c2 t2 giving σx2 (t) = x2 − x2 = (s2 + 2cx0 t + c2 t2 ) − (x0 + ct)2 = s2 − x20 = constant 11
One might be tempted to write P/2m in place of c, but it seems extravagant to introduce two constants where one will serve.
9
A simpler example: the “photon”
By extension of the same line of argument one can show (inductively) that the centered moments (x − x)n of all orders n are constant. Looking finally to the mean motion of the “correlation operator” C ≡ 12 (xp + p x) we find d dt C
=
1 2i [(xp
+ p x), cp ] = cp = cp
giving C = a + cpt The motion of the “correlation coefficient” xp + px C= − xp 2
(19)
can therefore be described C(t) = (a + cpt) − (x0 + ct)p = a − px0 The correlation coefficient C is, in other words, also constant. We conclude that for the “photonic system” σp2 (t)σx2 (t) = constant = P 2 · (s2 − x20 ) ≥ (/2)2 + (a − px0 )2
according to Schr¨ odinger
and on these grounds that the parameters {x0 , s, P, p and a} are subject to a constraint which can (compare (16)) be written P 2 (s2 − x20 ) − (a − px0 )2 ≥ (/2)2
(20)
From the constancy of the moments of principal interest to us we infer that the “photonic” system is non-dispersive. That same conclusion is supported also by this alternative line of argument: (17) gives rise to a “Schr¨ odinger equation” ∂ ∂ which can be written c( i ∂x )ψ = i ∂t ψ or more simply (∂x + 1c ∂t )ψ = 0 and the general solution of which is well known to move “rigidly” (which is to say: non-dispersively) to the right: ψ(x, t) = f (x − ct) Only at (20) does the quantum mechanical photonic system differ in any obvious respect from its classical counterpart. It seems to me curious that the system has not been discussed more widely. The system—which does not admit of Lagrangian formulation—derives some of its formal interest from the circumstance that both T-invariance and P-invariance are broken.
10
Status of Ehrenfest’s Theorem
4. Computational features of the general case. One could without difficulty—
though I on this occasion won’t—construct similarly detailed accounts of the momental dynamics of the systems free fall
:
H=
harmonic oscillator
:
H=
1 2m 1 2m
p2 + mg x p2 + mω 2 x2
and, indeed, of any system with a Hamiltonian H = c1 p2 + c2 (xp + px) + c3 x2 + c4 p + c5 x + c6 1 which depends at most quadratically upon the operators x and p. To illustrate problems presented in the more general case I look to the system H=
1 2m
p2 + 14 k x4
(21)
The classical equations of motion read
p˙ = −kx3 x˙ =
(22)
1 mp
while Ehrenfest’s relations (2) become d dt p d dt x
= −kx3 =
1 m p
(23.1)
The latter are, as Ehrenfest was the first to point out, exact corollaries of ∂ the Schr¨ odinger equation H|ψ) = i ∂t |ψ), and they are in an obvious sense “reminiscent” of their classical counterparts. But (23.1) does not provide an instance of (22), for the simple reason that x3 and x are distinct variables. More to the immediate point, (23.1) does not comprise a complete and soluable system of differential equations. In an effort to achieve “completeness” we look to 3 d dt x
=
3 1 i [x , H]
=
3 2 1 2mi [x , p ] 3 2
[x , p ] = [x3 , p ]p + p[x3 , p ] = 3i(x2 p + p x2 )
=
2 3 2m (x p
+ p x2 )
(23.2)
and discover that we must add (x2 p + p x2 ) to our list of variables. We look therefore to 2 2 2 2 d 1 dt (x p + p x ) = i [(x p + p x ), H] By tedious computation [(x2 p + p x2 ), p2 ] = i(xp2 + 2pxp + p2 x) [(x2 p + p x2 ), x4 ] = −8i x5
11
Momental hierarchy
so we have 2 d dt (x p
+ p x2 ) =
2 1 2m (xp
+ 2pxp + p2 x) − 83 kx5
(23.2)
but must now add both (xp2 + 2pxp + p2 x) and x5 to our list of variables. Pretty clearly (since H introduces factors faster that [x, p] = i 1 can kill them) equations (23) comprise only the leading members of an infinite system of coupled first-order linear (!) differential equations. Writing down such a system—quite apart from the circumstance that it may require an infinite supply of paper and ink—poses an algebraic problem of a high order, particularly in the more general case H=
1 2m
p2 +V (x) V (x) described by power series, or Laplace transform, or. . .
and especially in the most general case H = h(x, p). But assuming the system to have been written down, solving such a system poses a mathematical problem which is qualitatively quite distinct both from the problem of solving it’s (generally non-linear) classical counterpart p˙ = −V (x) : equivalently m¨ x = −V (x) 1 x˙ = m p and from solving the associated Schr¨ odinger equation 1 ∂ 2 ∂ + V (x) ψ(x, t) = i ∂t ψ(x, t) 2m i ∂x Distinct from and, we can anticipate, more difficult than. But while the computational utility of the “momental formulation of quantum mechanics” can be expected to be slight except in a few favorable cases, the formalism does by its mere existence pose some uncommon questions which would appear to merit consideration. 5. The momental hierarchy supported by an arbitrary observable. Let A refer to
an arbitrary observable. According to (1) d dt A
=
1 i [A, H]
1 Noting that if A and B are hermitian then [A, B] is antihermitian but i [A, B] is again hermitian (which is to say: an acceptable “observable”), let us agree to write
A0 ≡ A 1 [A, H] A1 ≡ i A2 ≡ .. . An+1 ≡
1 1 i [ i [A, H], H]
1 i [An , H]
≡
≡
1 2 i
1 n i
A, H2
A, Hn
:
n = 0, 1, 2, . . .
(24)
12
Status of Ehrenfest’s Theorem
The H-induced quantum motion of the “momental heirarchy supported by A” can be described d dt An
= An+1
:
n = 0, 1, 2, . . .
(25)
The heirarchy truncates at n = m if and only if it is the case that Am+1 = 0 (which entails An = 0 for all n > m); if and only if, that is to say, An is a constant of the motion. In such a circumstance one has d m+1 At = 0 dt which entails that At is a polynomial in t; specifically At =
m
n 1 n! An 0 t
(26)
n=0
Several instances of just such a situation have, in fact, already been encountered. For example: let H have the “photonic” structure (17), and let A be assigned 1 m the meaning m! x ; the resulting heirarchy truncates in after m steps: A0 ≡ A1 = A2 = A3 =
1 m m! x 1 c1 (m−1)! xm−1 1 c2 (m−2)! xm−2 1 c3 (m−3)! xm−3
.. . Am = cm 1 (a physically uninteresting constant of the motion) An = 0 for n > m In §3 we had occasion to study just such heirarchies in the cases m = 1 and m = 2, and were—for reasons now clear—led to polynomials in t. We developed there an interest also in the truncated heirarchy A0 ≡ 12 (xp + px) A1 = cp A2 = 0 Tractability of another sort attaches to heirarchies which, though not truncated, exhibit the property of cyclicity, which in its simplest manifestation means that Am = λ A0 for some λ and some least value of m. Then Am+q = λ Aq , A2m = λ2 A0 and d m dt
At = λAt
13
Momental hierarchy
which again yields to solution by elementary means: At = sum of exponentials involving the mth roots of λ For example: let H have the generic quadratic structure H = 12 ap2 + 12 b x2 and let A be assigned the meaning x (alternatively p); A0 ≡ p A1 = −b x A2 = −abp .. .
A0 ≡ x A1 = ap A2 = −bax .. .
Each of the preceding hierarchies is cyclic, with period 2 and λ = −ab. If we set a = 1/m and b = mω 2 then λ = −ω 2 , and we obtain results that bear on the quantum mechanics of a harmonic oscillator ; in particular, we have d 2 dt
xt = −ω 2 xt
which informs us that xt oscillates harmonically for all |ψ):12 the standard Gaussian assumption is superfluous. In the limit ω ↓ 0 (i.e., for b = 0) the preceeding heirarchies (instead of being cyclic) truncate, and we obtain results appropriate to the quantum mechanics of a free particle. When A is assigned the meaning x2 (alternatively p2 ) we are led to hierarchies A0 ≡ x2
A0 ≡ p 2
A1 = a(xp + px) .. .
A1 = −b(xp + px) .. .
which become identical to within a factor at the second step, and it is thereafter that the hierarchy continues cyclically A0 ≡ 12 (xp + px) A1 = (ap2 − b x2 ) A2 = −2ab(xp + px) .. . d with period 2 and λ = −4ab. From the fact that dt x2 t is oscillatory it follows that x2 t = constant + oscillatory part 12
This seldom remarked fact was first brought casually to my attention years ago by Richard Crandall.
14
Status of Ehrenfest’s Theorem
and from this we conclude it to be a property of harmonic oscillators that (for all |ψ)) σx2 (t) and σp2 (t) “ripple” with twice the base frequency of the oscillator . Hierarchies into which 0 intrudes are necessarily truncated, and those which contain a repeated element are necessary cyclic, but in general one can expect a hierarchy to be neither truncated nor cyclic. In the general case one has At =
∞
n 1 n! An 0 t
within some radius of convergence
(27)
n=0
which does give back (26) when the hierarchy truncates, does sum up nicely in cyclic cases,13 and can be construed to be a generating function for the expectation values of the members of the hierarchy. This, however, becomes a potentially useful point of view only if one (from what source?) has independent knowledge of At . I note in passing that at (27) we have recovered a result which is actually standard; in the Heisenberg picture one writes A t = e− i Ht A 0 e+ i Ht 1
1
to describe quantum motion, and makes use of the operator identity = ≡
∞ n=0 ∞
1 1 n A , Hn tn n! i n 1 n! A n t
(28)
n=0
from which (27) can be recovered as an immediate corollary. 6. Reconstruction of the wave function from momental data.14 While A is a
“moment” in the sense that it describes the expected mean (1st moment) of a set of A-measurements, I propose henceforth to reserve for that term a more restrictive meaning. I propose to call the numbers xn —which by standard usage are the moments of probability distribution |ψ ∗ (x)ψ(x)|—“moments of the wave function,” though the wave function ψ(x) is by nature a probability amplitude. In that extended sense, so also are the numbers pn “moments of the wave function.” But so also are some other numbers. My assignment is to describe the least population of such numbers sufficient to the purpose at hand (reconstruction of the wave function), and then to show how they in fact achieve that objective. It proves convenient to consider those problems in reverse order, and to begin with review of some classical probability theory:
Note that while An = 0 implies truncation, the |ψ)-dependent circumstance An = 0 does not; similarly, An = λA0 implies cyclicity but An = λAn does not. 14 Time is passive in the following discussion (all I have to say should be understood to hold at each moment), so allusions to t will be dropped from my notation. 13
15
Reconstruction of the wave function
Let P (x, p) be some bivariate distribution function. The marginal moments xm and pn can be described in terms of the associated marginal distribution functions f (x) ≡ P (x, p)dp and g(p) ≡ P (x, p)dx m m n x = x f (x)dx and p = pn g(p)dp If x and p are statistically independent random variables then P (x, p) contains no information not already present in f (x) and g(p); indeed, one has P (x, p) = f (x)g(p)
giving xm pn = xm pn : x and p independent
But that is a very special situation; the general expectation must be that x and p are statistically dependent. Then P (x, p) contains information not present in f (x) and g(p), the mixed moments xm pn individually contain information not present within the set of marginal moments, and one must be content to write xm pn = xm pn P (x, p)dxdp That f (x) can be reconstructed from the data {xm : m = 0, 1, 2, . . .}, and g(p) from the data {pn : n = 0, 1, 2, . . .}, has in effect been remarked already in §1; form F (β) ≡
∞
1 m!
xm
m i βx = e = βx
i
i
e βx f (x)dx
m=0
Then
f (x) =
1 h
g(p) =
1 h
e− βx F (β)dβ i
and similarly
e− αp G(α)dα i
i where G(α) ≡ e αp . The question now arises: How (if at all) can one reconstruct P (x, p) from the data {xm pn }? The answer is: By straightforward extension of the procedure just described. Group the mixed moments according to their net order 1 x x2 x3
p
xp
x2 p
p2
xp2 .. .
p3
16
Status of Ehrenfest’s Theorem
and form Q(α, β) ≡ =
∞ k=0 ∞
1 i k k!
k
k n
xn pk−n β n αk−n
n=0
1 i k (αp k!
i + βx)k = e (αp+βx)
k=0
=
i
e (αp+βx) P (x, p)dxdp
Immediately P (x, p) =
1 h2
i i e− (αp+βx) e (αp+βx) dqdy moment data xm pn resides here
(29)
i i i If x and p are statistically independent, then e (αp+βx) = e αp e βx and we recover P (x, p) = f (x)g(p). My present objective is to construct the quantum counterpart of the preceding material, and for that purpose the so-called “phase space formulation of quantum mechanics” provides the natural tool. This lovely theory, though it has been available for nearly half a century,15 remains—except to specialists in quantum optics16 and a few other fields—much less well known than it deserves to be. I digress now, therefore, to review its relevant essentials: Seeds of the theory were planted by Hermann Weyl (see Chapter IV, §14 in his Gruppentheorie und Quantenmechanik (2nd edition, )) and Eugene Wigner: “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749 (1932). Those separately motivated ideas were fused and systematically elaborated in a classic paper by J. E. Moyal (who worked in collaboration with the British statistician M. E. Bartlett): “Quantum mechanics as a statistical theory,” Proc. Camb. Phil. Soc. 45, 92 (1949). The foundations of the subject were further elaborated in the ’s by T. Takabayasi (“The formulation of quantum mechanics in terms of ensembles in phase space,” Prog. Theo. Phys. 11, 341 (1954)), G. A. Baker Jr. (“Formulation of quantum mechanics in terms of the quasi-probability distribution induced on phase space,” Phys. Rev. 109, 2198 (1958)) and others. For a fairly detailed account of the theory and many additional references, see my quantum mechanics (), Chapter 3, pp. 27–32 and pp. 99 et seq. or the Reed College thesis of Thomas Banks: “The phase space formulation of quantum mechanics” (1969). 16 L. Mandel & E. Wolf, in Optical coherence and quantum optics (), make only passing reference (at p. 541) to the phase space formalism. But Mark Beck (see below) has supplied these references: Ulf Leonhardt, Measuring the Quantum State of Light (); M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121 (1984). 15
17
Reconstruction of the wave function
To the question “What is the self-adjoint operator A that should, for the purposes of quantum mechanical application, be associated with the classical observable A(x, p)?” a variety of answers have been proposed.17 The rule of association (or correspondence procedure) advocated by Weyl can be described i A(x, p) = a(α, β)e (αp+βx) dαdβ | | weyl transformation ↓ i A= a(α, β)e (α p+β x) dαdβ
(30)
i
It was to the wonderful properties of the operators E(α, β) ≡ e (α p+β x) that Weyl sought to draw attention.18 Those entail in particular that if A ←−−−−− A(x, p)
and
Weyl
B ←−−−−− B(x, p) Weyl
then trace AB =
1 h
A(x, p)B(x, p)dxdp
and permit this description of the inverse Weyl transformation: i + 1 A −−−−−→ A(x, p) = trace AE (α, β) e (αp+βx) dαdβ h Weyl
(31)
(32)
Those facts acquire their relevance from the following observations: familiarly, A = (ψ|A|ψ) can be written A = trace Aρρ ρ ≡ |ψ)(ψ| is the density matrix associated with the state |ψ) Writing A −−−−−→ A(x, p) Weyl
and
ρ −−−−−→ h · Pψ (x, p) Weyl
one therefore has A =
A(x, p)Pψ (x, p)dxdp
17
(33)
See J. R. Shewell, “On the formation of quantum mechanical operators,” AJP 27, 16 (1959). 18 Among those many wonderful properties are the trace-wise orthonormality property ¯ = hδ(α − α ¯ trace E(α, β)E+ (¯ α, β) ¯ )δ(β − β) from which it follows in particular that trace E(α, β) = hδ(α)δ(β)
18
Status of Ehrenfest’s Theorem
where by application of (32) we have i i Pψ (x, p) = h12 (ψ|e− (α p+β x) |ψ)e (αp+βx) dαdβ
(34)
which can by fairly quick calculation be brought to the form i 2 Pψ (x, p) = h ψ ∗ (x + ξ) e2 p ξ ψ(x − ξ)dξ
(35)
At (35) we have recovered the famous “Wigner distribution function,” which Wigner in was content simply to pluck from his hat.19 The function Pψ (x, p)—which in the phrase space formalism serves to describe the “state” of the quantum system, but is invariable real-valued—possesses many of the properties one associates with the term “distribution function; ” one finds, for example, that
Pψ (x, p)dxdp = 1 Pψ (x, p)dp = |ψ(x)|2 and Pψ (x, p)dx = |ϕ(p)|2
where ϕ(p) ≡ (p|ψ) is the Fourier transform of ψ(x) ≡ (x|ψ). And even more to the point: the Wigner distribution enters at (33) into an equation which is formally identical to the equation used to define the expectation value A(x, p) in classical (statistical) mechanics. But Pψ (x, p) possess also some “weird” properties—properties which serve to encapsulate important respects in which quantum statistics is non-standard, quantum mechanics non-classical Pψ (x, p) is not precluded from assuming negative values Pψ (x, p) is bounded : |Pψ (x, p)| ≤ 2/h Pψ (x, p) = |ψ(x)|2 · |ϕ(p)|2 is impossible and for those reasons (particularly the former) is called a “quasi-distribution” by some fastidious authors. From the marginal moments {xn : n = 0, 1, 2, . . .} it is possible (by the classical technique already described) to reconstruct |ψ(x)|2 but not the wave function ψ(x) itself , for the data set contains no phase information. A similar remark pertains to the reconstruction of ϕ(p) from {pm : m = 0, 1, 2, . . .}. But ψ(x) and its “Wigner transform” Pψ (x, p) are equivalent objects in the sense that they contain identical stores of information; from Pψ (x, p) it is possible to recover ψ(x), by a technique which I learned from Mark Beck20 and will 19
Or perhaps from the hat of Leo Szilard; in a footnote Wigner reports that “This expression was found by L. Szilard and the present author some years ago for another purpose,” but cites no reference. 20 Private communication. Mark does not claim to have himself invented the trick in question, but it was, so far as I am aware, unknown to the founding fathers of this field.
19
Reconstruction of the wave function
describe in a moment. The importance (for us) of this fact lies in the following observation: Momental data sufficient to determine Pψ (x, p) is sufficient also to determine ψ(x), to with an unphysical over-all phase factor. The construction of ψ(x) ←−−−−− Pψ (x, p) (Beck’s trick) proceeds as follows: Wigner
By Fourier transformation of (35) obtain i ˆ ˆ Pψ (x, p)e−2 p ξ dp = ψ ∗ (x + ξ) δ(ξ − ξ)ψ(x − ξ)dξ ˆ ψ(x − ξ) ˆ = ψ ∗ (x + ξ)
Select a point a at which Pψ (a, p) dp = ψ ∗ (a) ψ(a) = 0.21 Set ξˆ = a − x to obtain i Pψ (x, p)e−2 p(a−x) dp = ψ ∗ (a) ψ(2x − a) and by notational adjustment 2x − a → x obtain i p (x−a) dp ψ(x) = [ψ ∗ (a)]–1 · Pψ ( x+a 2 , p) e ↓ = [ψ ∗ (0)]–1 ·
(36)
i
Pψ ( x2 , p) e p x dp in the special case a = 0
where the prefactor is, in effect, a normalization constant, fixed to within an arbitrary phase factor. Returning to the question which originally motivated this discussion— What least set of momental data is sufficient to determine the state of the quantum system?—we are in position now to recognize that an answer was implicit already in (34), which (taking advantage of the reality of the Wigner distribution, and in order to regain contact with notations used by Moyal) I find it convenient at this point to reexpress i 1 Pψ (x, p) = h2 Mψ (α, β)e− (αp+βx) dαdβ (38) i
Mψ (α, β) ≡ (ψ|e (α p+β x) |ψ) = (ψ|E(α, β)|ψ)
(39)
Evidently E(α, β) =
∞ k=0
1 i k k!
k
Mk−n,n αk−n β n
n=0
Mm,n ≡
(40) m p -factors and n x-factors
(41)
all orderings
= sum of
m+n n
terms altogether
Such a point is, by ψ ∗ (x) ψ(x) dx = 1, certain to exist. It is often most convenient (but not always possible) to—with Beck—set a = 0. 21
20
Status of Ehrenfest’s Theorem
so it is the momental set {Mn,m } that provides the answer to our question. Looking now in more detail to the primitive operators Mm,n , the operators of low order can be written M0,0 = 1 M1,0 = p M0,1 = x M2,0 = pp M1,1 = px + xp M0,2 = xx M3,0 = ppp M2,1 = ppx + pxp + xpp M1,2 = px x + xpx + x xp M0,3 = x x x .. . It is hardly surprising—yet not entirely obvious—that 1 number of terms Mm,n
←−−−−−−−− pm xn
(42)
Weyl
I say “not entirely obvious” because by original definition A ←−−−−− A(x, p) Weyl
assumes A(x, p) to be Fourier transformable, which polynomials are not.22 I digress now to indicate how by natural extension the Weyl transform acquires its surprising robustness and utility. Any operator presented in the form A = sum of powers of x and p operators can, by virtue of the fundamental commutation relation, be written in many ways. In particular, any A can, by repeated use of xp = p x + i1, be brought to “px-ordered form” (else “x p-ordered form”) in which all p-operators stand left of all x-operators (else the reverse). I find it convenient to write (idiosyncratically) x F (x, p) p ≡ result of x p-ordered substitution into F (x, p) (43) p F (x, p) x ≡ result of p x-ordered substitution into F (x, p) 22
On the other hand, that definition —(30)—is built upon an assertion i
i
e (α p+β x) ←−−−−−−−− e (αp+βx) Weyl
from which (42) appears to follow as an immediate implication.
21
Reconstruction of the wave function
and, inversely, to let Fxp (x, p) denote the function which yields F by x p-ordered substitution: F = x Fxp (x, p) p = p Fpx (x, p) x
(44) :
reverse-ordered companion of the above
For example, if F ≡ xpx = x2 p − i x = p x2 + i x then Fxp (x, p) = x2 p − ix but Fpx (x, p) = x2 p + ix Some sense of (at least one source of) the frequently great computational utility of “ordered display” can be gained from the observation that (x|F|y) = (x|F|p)dp(p|y) : mixed representation trick i 1 √ = h Fxp (x, p)e− py dp (45) = (x|p)dp(p|F|y) i 1 √ = h e+ xp Fpx (y, p) dp One of the principal recommendations of Weyl’s procedure is that it lends itself so efficiently to the analysis of operator ordering/re-ordering problems; if A and B commute with their commutator (as, in particular, x and p do) then23 eA + B = e+ 2 [ A, B] · eB eA = e− 2 [ A, B] · eA eB 1
1
which entail i
e (αp +β x) =
1i i i e+ 2 αβ · e β x e αp
− 12
e
i αβ
·e
i αp
e
i βx
:
x p-ordered display (46)
:
p x-ordered display
Returning with this information to (30) we have 23
The following are among the most widely known of the identities which issue from “Campbell-Baker-Hausdorff theory,” which originates in the prequantum mechanical mathematical work of J. E. Campbell (), H. F. Baker () and F. Hausdorff (), but attracted wide interest only after the invention of quantum mechanics. For a good review and references to the classical literature, see R. M. Wilcox, “Exponential operators and parameter differentiation in quantum mechanics,” J. Math. Phys. 8, 962 (1967). Or “An operator ordering technique with quantum mechanical applications” () in my collected seminars.
22
Status of Ehrenfest’s Theorem
A(x, p) =
i
a(α, β)e (αp+βx) dαdβ
↑ | Weyl ↓ i A= a(α, β)e (α p+β x) dαdβ 1 i i i = a(α, β)e+ 2 αβ · e β x e αp dαdβ ∂2 = x exp 12 i ∂x∂p A(x, p) p from which we learn that Axp (x, p) = exp + Apx (x, p) = exp −
1 ∂2 2 i ∂x∂p 1 ∂2 2 i ∂x∂p
A(x, p)
A(x, p)
(47)
Suppose (which is to revisit a previous example) we were to take A(x, p) = px2 ; then (47) asserts A(x, p) ≡ px2 −−−−−−−−→ A = x2 p − i x Weyl
= p2 x + i x while by explicit calculation we find = xpx = 13 (px x + xp x + x xp) ≡ 13 M1,2 Here we have brought patterned order and efficiency to a calculation which formerly lacked those qualities, and have at the same time shown how the Weyl correspondence comes to be applicable to polynomial expressions. 7. A shift of emphasis—from moments to their generating function. We began
with an interest—Ehrenfest’s interest—in (the quantum dynamical motion of) only a pair of moments (x and p), but in consequence of the structure of (2) found that a collateral interest in mixed moments of all orders was thrust upon us. Here I explore implications of some commonplace wisdom: When one has interest in properties of an infinite set of objects, it is often simplest and most illuminating to look not to the objects individually but to their generating function. I look now, therefore, in closer detail to properties of a function which we have already encountered—to what I call the “Moyal function” i
Mψ (α, β) ≡ (ψ|e (α p+β x) |ψ) = (ψ|E(α, β)|ψ) = E(α, β) with E(α, β) unitary
(48)
23
Momental generating function
which was seen at (38) to be precisely the Fourier transform of the Wigner distribution, and therefore to be (by performance of Beck’s trick) a repository of all the information borne by |ψ). To describe the motion of all mixed moments at once we examine the time derivative of Mψ (α, β), which by (1) can be described ∂ ∂t Mψ (α, β)
=
1 i [E(α, β),H]
(49)
Proceeding on the assumption that H ←−−−−− H(x, p) = Weyl
˜ ˜ βx) ˜ i (αp+ h(˜ α, β)e dα ˜ dβ˜
we have ∂ ∂t Mψ (α, β)
=
1 i
˜ ˜ h(˜ α, β)[E(α, β), E(˜ α, β)]d α ˜ dβ˜
But it is24 an implication of (46) that ˜ = (eϑ − e−ϑ ) ·E(α + α ˜ [E(α, β), E(˜ α, β)] ˜ , β + β) = 2i sin ϑ : ϑ ≡
(50)
1 ˜ 2 (αβ −
βα ˜)
so we can write ∂ ∂t Mψ (α, β)
=
2
=
2
˜ sin h(˜ α, β)
=
αβ−β ˜ α ˜ 2
h(˜ α − α, β˜ − β) sin
˜ α ˜ , β + β)d ¯ dβ¯ · Mψ (α + α
αβ−β ˜ α ˜ 2
˜ α α, β)d ˜ dβ˜ · Mψ (˜
˜ · Mψ (˜ ˜ α T(α, β; α ˜ , β) α, β)d ˜ dβ˜ ˜ ≡ 2 h(˜ T(α, β; α ˜ , β) α − α, β˜ − β) sin
(51.1) αβ−β ˜ α ˜ 2
(51.2)
Equation (51.1)—which formally resembles (and is ultimately equivalent to) this formulation of Schr¨ odinger equation ∂ ∂t (x|ψ)
=
(x|H|˜ x)d˜ x(˜ x|ψ)
—is, in effect, a giant system of coupled first-order differential equations in the mixed moments of all orders; it asserts that the time derivatives of those moments are linear combinations of their instantaneous values, and that it is the responsibility of the Hamiltonian to answer the question “What linear combinations?” and thus to distinguish one dynamical system from another. 24
See Chapter 3, p. 112 of quantum mechanics () for the detailed argument.
24
Status of Ehrenfest’s Theorem
By Fourier transformation one at length24 recovers
∂ ∂ ∂ 2 ∂ ∂ H(x, p)Pψ (x, p) P (x, p) = sin − (52) ψ ∂t 2 ∂x H ∂p P ∂x P ∂p H ∂H ∂ ∂ 2 = ∂x ∂p − ∂H ∂p ∂x Pψ (x, p) +“quantum corrections” of order O( ) Poisson bracket [H, Pψ ] which is the “phase space formulation of Schr¨ odinger’s equation” in its most frequently encountered form. Equation (52) makes latent good sense in all cases H(x, p), and explicit good sense in simple cases; for example: in the “photonic case” H = cp (see again §3) one obtains ∂ ∂t Pψ (x, p)
∂ = −c ∂x Pψ (x, p)
1 2 while for an oscillator H = 2m p + 12 mω 2 x2 we find 2 ∂ ∂ 1 ∂ ∂t Pψ (x, p) = mω x ∂p − m p ∂x Pψ (x, p) ↓ 1 ∂ = −m p ∂x Pψ (x, p)
in the “free particle limit” ω ↓ 0
(53.1)
(53.2) (53.3)
I postpone discussion of the solutions of those equations (but draw immediate attention to the fact that each of those cases is so quadratically simple that “quantum corrections” are entirely absent). . . in order to draw attention to my immediate point, which is that in each of those cases (51) is meaningless, for the simple reason that none of those Hamiltonians is Fourier transformable; in each case h(α, β) fails to exist. In a first effort to work around this problem, let us back up to (49) and consider again the case H = cp: then ∂ ∂t Mψ (α, β)
1 = c i [E(α, β), p ]
(54)
It is an implication of (50) that i
¯ [E(α, β), e αp ]=
∞
k 1 i ¯ [E(α, β), pk k! α
k=0
= 2i sin
−β α¯ 2
]=0+α ¯ · i [E(α, β), p ] + · · ·
E(α + α ¯ , β) = α ¯ · − i β E(α, β) + · · ·
from which we infer [E(α, β), p ] = −β E(α, β)
(55)
Returning with this information to (54) we have ∂ ∂t Mψ (α, β)
1 = − i cβMψ (α, β)
(56)
˜ = − 1 cδ(˜ ˜ which can be cast in the form (51.1) with T(α, β; α ˜ , β) α −α)δ(β˜ −β)β. i ˜ The implication appears to be that we should in general expect T(α, β; α ˜ , β) to have the character not of a function but of a distribution.
25
Momental generating function
It is in preparation for discussion of less trivial cases (free particle and oscillator) that I digress now to explore some consequences of the identity (55),25 which can be written E(α, β)p = (p − β1)E(α, β) or again as the “shift rule” (most familiar in the case α = 0) E(α, β)pE –1 (α, β) = (p − β1) Immediately E(α, β)pm E –1 (α, β) = (p − β1)m or again [E(α, β), pm ] = {(p − β1)m − pm } E(α, β) which—because {etc.} introduces “dangling p-operators” except in the cases m = 0 and m = 1—does, as it stands, not quite serve our purposes. It is, however, an implication of (46) that
∂ m E(α, β) i ∂α
= (p − 12 β1)m E(α, β)
and therefore that
∂ i ∂α ∂ i ∂α
− 12 β + 12 β
m m
E(α, β) = (p − β1)m E(α, β) p m E(α, β)
E(α, β) =
So we obtain [E(α, β), pm ] =
∂ i ∂α
=
− 12 β
m
−
∂ i ∂α
0 −β E(α, β)
∂ E(α, β) −2 i β ∂α
+ 12 β
m
:
m=0
:
m=1
:
m=2
E(α, β)
(57.1)
.. .
and, by similar argument,26 [E(α, β), xn ] = 25
∂ i ∂β
+ 12 α
n
−
∂ i ∂β
− 12 α
n
E(α, β)
(57.2)
We want—minimally—to be in position to say useful things about the commutators [E(α, β), p2 ] and [E(α, β), x2 ]. 26 It is simpler to make substitions p → +x, x → −p, α → +β, β → −α (which by design preserve both [x, p ] and the definition of E(α, β)) into the results already in hand.
26
Status of Ehrenfest’s Theorem
Returning with this information to the case of an oscillator, we have ∂ ∂t Mψ (α, β)
= = =
1
↓ =
2 2 2 1 1 1 i 2m [E(α, β), p ] + 2 mω [E(α, β), x ] 2 1 1 ∂ 1 ∂ + 2 i α ∂β E(α, β) i 2m − 2 i β ∂α + 2 mω ∂ m β ∂α
∂ Mψ (α, β) − mω 2 α ∂β
1 ∂ m β ∂α Mψ (α, β)
in the “free particle limit”
(58.1) (58.2)
Equations (56) and (58) are as simple as—and bear a striking resemblance to— their Wignerian counterparts (53). But to render (58.2)—say—into the form ˜ = − 1 δ (˜ ˜ in conformity (51) we would have to set T(α, β; α ˜ , β) α − α)δ(β˜ − β)β, m with our earlier conclusion concerning the generally distribution-like character ˜ The absence of -factors on the right sides of (58) is of the kernel T(α, β; α ˜ , β). consonant with the absence of “quantum corrections” on the right sides of (53), but makes a little surprising the (dimensionally enforced) 1/i that appears on the right side of (56). One could but I won’t. . . undertake now to describe the 1 2 analogs of (58) which arise from H(x, p) = 2m p + V (x) and from Hamiltonians of still more general structure. Instead, I take this opportunity to underscore what has been accomlished at (58.1). By explicit expansion of the expression on the left we have ∂ ∂t Mψ (α, β)
=
∂ ∂t
1 + i αp + βx 2 2 2 + 12 i α p + αβp x + x p + β 2 x2 + · · ·
while expansion of the expression on the right gives 1
2 ∂ ∂ m β ∂α − mω α ∂β Mψ (α, β) 1 = i m βp − mω 2 αx 1 2 2 1 2 + 12 i m 2αβp + m β
− mω 2 α2 p x + xp − mω 2 2αβx2 + · · ·
Term-by-term identification gives rise to a system of equations: α1
:
β1
:
2
:
αβ
:
2
: .. .
α β
2 d dt p = −mω x d 1 dt x = m p 2 2 d dt p = −mω xp + p x 2 2 2 d 2 dt x p + p x = m p − 2mω x 2 d 1 dt x = m xp + p x
27
Solution of Moyal’s equation
which in the “free particle limit” become α1
:
β1
:
2
:
αβ
:
2
: .. .
α β
d dt p = 0 d 1 dt x = m p 2 d dt p = 0 2 d 2 dt x p + p x = m p 2 d 1 dt x = m xp + p x
These are precisely the results achieved in §2 by other means. It seems, on the basis of such computation, fair to assert that equations of type (58) provide a succinct expression of Ehrenfest’s theorem in its most general form.27 Let us agree, in the absence of any standard terminology, to call (52) the “Wigner equation,” and its Fourier transform—the generalizations of (56)/(58) —the “Moyal equation.” Evidently solution of Moyal’s equation—a single partial differential equation—is equivalent to (though poses a very different mathematical problem from) the solution of the coupled systems of ordinary differential “moment equation” of the sort anticipated in §4 and encountered just above.28 In the next section I look to the. . . 8. Solution of Moyal’s equation in some representative cases. Look first to the
“photonic system” H(x, p) = cp. Solutions of the Wigner equation (53.1) can be described ∂ Pψ (x, p; t) = exp − ct ∂x Pψ (x, p; 0) = Pψ (x − ct, p; 0)
(59.1)
while the associated Moyal equation (56) promptly yields i
Mψ (α, β; t) = e cβt · Mψ (α, β; 0) 27
(59.2)
It should in this connection be observed that the equations to which we have been led, though rooted in formalism based upon the Weyl correspondence, have in the end a stand-alone validity, and are therefore released from the criticism that there exist plausible alternatives to Weyl’s rule (see again the paper by J. L. Shewell to which I made reference in footnote 17), and that its adoption is in some sense an arbitrary act. A similar remark pertains to other essential features of the phase space formalism. 28 One is reminded in this connection of the partial differential wave equation that arises by a “refinement procedure” from the system of ordinary differential equations that describe the motion of a discrete lattice. And it becomes in this light natural to ask: “Does Moyal’s equation admit of representation as the field equation implicit in some Lagrange density? Does it provide, on other words, an instance of a Lagrangian field theory?”
28
Status of Ehrenfest’s Theorem
These simple results are simply interrelated—if (compare (38)) i 1 Pψ (x, p; 0) = h2 Mψ (α, β; 0)e− (αp+βx) dαdβ then (59.2) immediately entails (59.1)—but cast no light on a fundamental question which I must for the moment be content to set aside: What general constraints/side conditions does theory impose upon the functions Pψ (x, p; 0) and Mψ (α, β; 0)? Looking next to the oscillator: equations (53.2) and (58.1), which have already been remarked to “bear a striking resemblance to” one another, are in fact structurally identical; whether one proceeds by notational adjustment {x → u, +p/mω → v} from (53.2) {α → u, −β/mω → v} from (58.1) one obtains an equation of the form ∂ ∂t F (u, v)
∂ ∂ = ω u ∂v − v ∂u F (u, v)
The differential operator within braces is familiar from angular momentum theory as the generator of rotation on the (u, v)-plane; immediately ∂ ∂ F (u, v; t) = exp u ∂v ωt F (u, v; 0) − v ∂u = F (u cos ωt − v sin ωt, u sin ωt + v cos ωt; 0) —the accuracy of which can be confirmed by quick calculation. So we have Pψ (x, p; t) = Pψ (x cos ωt−(p/mω) sin ωt, mωx sin ωt+p cos ωt; 0)
(60.1)
which, though entirely and accurately quantum mechanical in its meaning, conforms well to the familiar classical fact that Hoscillator (x, p) generates synchronous elliptical circulation on the phase plane. Similarly (or by Fourier transformation) Mψ (α, β; t) = Mψ (α cos ωt+(β/mω) sin ωt, −mωα sin ωt+β cos ωt; 0)
(60.2)
according to which the circulation on the (α, β)-plane is relatively retrograde— as one expects it to be.29 Information concerning the time-dependence of the nth -order moments can now be extracted from (αp + β x)n t = ([α cos ωt+(β/mω) sin ωt] p + [−mωα sin ωt+β cos ωt] x)n 0 (61) 29
The simple source of that expectation: If x = xP (x) dx then xP (x + a) dx = x − a : compare the signs!
29
Solution of Moyal’s equation
Evidently and remarkably, the nth -order moments move among themselves — independently of any reference to the motion of moments of any other order.And Fourier analysis of their motion will (consistently with a property of σx2 (t) reported in §5, and in consequence ultimately of De Moive’s theorem) reveal terms of frequencies ω, 2ω, 3ω, . . . nω. When one attempts to bring patterned computational order to the detailed implications of (61)—which, I repeat, was obtained by solution of Moyal’s equation in the oscillatory case—one is led spontaneously to the reinvention of some standard apparatus. It is natural to attempt to display “synchronous elliptical circulation on the phase plane” as simple phase advancement on a suitably constructed complex plane—natural therefore to notice that the dimensionless construction 1 (αp + β x) can be displayed 1 (αp
a + bbb + β x) = aa
provided the dimensionless objects on the right are defined a ≡ a1 + ia2 ≡
mω 2
1 α + i √2mω β
b ≡ a1 − ia2 ≡ a∗ a2 ≡ a ≡ a1 + ia
√ 1 p 2mω +
− i mω 2 x
a2 ≡ a b ≡ a1 − ia
The motion (elliptical circulation) of α and β α −→ α cos ωt + (β/mω) sin ωt β −→ −mωα sin ωt + β cos ωt becomes in this notation very easy to describe a −→ ae−iωt and so also, therefore, does the motion of (αp + β x)n ; we have
a + bbb)n (aa
t
= (ae−iωt a + be+iωt b)n 0
(62)
which, by the way, shows very clearly where the higher frequency components come from. But this is in (reassuring) fact very old news, for a and b are familiar as the ↓ and ↑ “ladder operators” described by Dirac in §34 of his Principles of Quantum Mechanics; they have the property that a, b ] = 1 [a and permit the oscillator Hamiltonian to be described H = ω b a + 12 1
30
Status of Ehrenfest’s Theorem
Working in the Heisenberg picture, one therefore has a˙ =
1 a i [a , H]
= −iω a giving a(t) = e−iωt a(0) b (t) = e+iωt b(0)
of which (?) can be considered a corollary. It is interesting to notice, pursuant to a previous remark concerning higher frequency components, that if A ≡ product of m b -factors and n a -factors in any order then
A˙ = i(m − n)ωA
giving A(t) = ei(m−n)ωt A(0)
It is, in short, quite easy to obtain detailed information about how the motion of all numbers of the type A. But only exceptionally are such numbers of direct physical interest, since only exceptionally is A hermitian (representative of an observable), and the extraction of information concerning the motion of (x, p)-moments can be algebraically quite tedious. Quantum opticians (among others) have, however, stressed the general theoretical utility, in connection with many of the questions that arise from the phase space formalism, of operators imitative of a and b. In the “free particle limit” equations (60) read Pψ (x, p; t) = Pψ (x − Mψ (α, β; t) = Mψ (α
1 m pt, p; 0) 1 +m βt, β; 0)
(63.1) (63.2)
Verification that (63.1) does in fact satisfy the “free particle Wigner equation” (53.3), and that (63.2) does satisfy the associated Moyal equation (58.2), is too immediate to write out. From the latter one obtains (compare (61))
(αp + β x)n
t
= ([α +
1 m βt]p
+ β x)n
0
(64)
which provides an elegantly succinct summary of material developed by clumbsy means in §2. But the definitions of a and b become, in this limit, meaningless; that fact touches obliquely on the reason that I found it simplest to treat the oscillator first. 9. When is Pψ (x, p) a “possible” Wigner function? When, within standard
∂ |ψ) we usually—and when we write quantum mechanics, we write H|ψ) = i ∂t A = (ψ|A|ψ) we invariably—understand |ψ) to be subject to the side condition (ψ|ψ) = 1. That condition is universal, rooted in the interpretive foundations of the theory.30 My present objective—responsive to a question posed already in connection with (59)—is to describe conditions which attach with similar 30
I will not concern myself here with the boundary, differentiability, continuity, single-valuedness and other conditions which which in individual problems attach typically and so consequentially to (for example) ψ(x) ≡ (x|ψ).
31
Side conditions to which Wigner & Moyal functions are subject
universality to the functions Pψ (x, p) and Mψ (α, β), and which collectively serve to distinguish admissible functions from “impossible” ones. The issue is made relatively more interesting by the circumstance that the Wigner distribution provides a representation of the “density matrix” ρ , and ρ embodies a richer concept of “state” than does |ψ). In this sense: |ψ) refers to the state of an individual system, while ρ refers to the state of a statistically described ensemble of systems.31 We imagine it to be the case32 that systems drawn from such an ensemble will be33 in state |ψ1 ) with probability p1 in state |ψ2 ) with probability p2 .. . in state |ψk ) with probability pk .. . Under such circumstances we expect to write A = pk (ψk |A|ψk )
(65.1)
k
= ordinary mean of the quantum means to describe the expected mean of a series of A-measurements. Exceptionally— when all members of the ensemble are in the same state |ψ)—the “ordinary” aspect of the averaging process is rendered moot, and we have = 0 + 0 + · · · + 0 + (ψ|A|ψ) + 0 + · · ·
(65.2)
It is of this “pure case” (the alternative, and more general, case being the “mixed case”) that quantum mechanics standardly speaks. Density matrix theory springs from the elementary observation that (65.1) can be expressed A = trace Aρρ ρ≡
(66) |ψk )pk (ψk |
k
pk are non-negative, subject to the constraint 31
(67) !
pk = 1
Commonly one omits all pedantic reference to an “ensemble,” and speaks as though simply uncertain of what state the system is actually in; “It might be in that state, but is more likely to be in this state. . . ” 32 But under what circumstances, and on what observational grounds, could we establish it to be the case? 33 Found to be? How? Quantum mechanics itself keeps fuzzing up the idea at issue, simple though it appears at first sight to be. And fuzzy language, though difficult to avoid, only compounds the problem. It is for this reason that I am inclined to take exception to the locution “measuring the quantum state” (to be distinguished from “preparing the quantum state”?) which has recently become fashionable, and is used even in the title of one of the publications cited in footnote 15.
32
Status of Ehrenfest’s Theorem
from which (65.2) can be recovered as a specialized instance. The operator ρ is hermitian;34 we are assured therefore that its eigenvalues ρk are real and mistake to confuse ρk its eigenstates |ρk ) orthogonal. It is, however, usually a! with pk , |ρk ) with |ψk ), the spectral representation ρ = |ρk )ρk (ρk | of ρ with (67); those associations can be made if and only if the states |ψk ) present in the ensemble are orthogonal , which may be the case,35 and by many authors is casually assumed to be the case,36 but in general such an assumption would do violence to the physics. Let us supppose—in order to keep the notation as simple as possible, and the computation as explicitly detailed—that our ensemble contains a mixture of only two states: ρ = |ψ)p(ψ| + |φ)q(φ| with p + q = 1 Operators of the construction Pψ ≡ |ψ)(ψ| are hermitian projection operators: Pψ2 = Pψ . Specifically, Pψ projects |α) −→ (ψ|α) · |ψ) onto the one-dimensional subspace (or “ray”) in state space which contains |ψ) as its normalized element. Generally trace (projection operator) = dimension of space onto which it projects ! ! so the calculation trace Pψ = (n|ψ)(ψ|n) = (ψ| |n)(n| |ψ) = (ψ|ψ) = 1 yields a result which might, in fact, have been anticipated, and puts us in position to write ρ = p Pψ + q Pφ ↓ trace ρ = p + q = 1
:
all cases
(68)
More informatively, ρ2 = p2 Pψ + q 2 Pφ + pq(Pψ Pφ + Pφ Pψ ) ↓ 2
trace ρ = p2 + q 2 + 2pq (ψ|φ)(φ|ψ) 0 ≤ (ψ|φ)(φ|ψ) ≤ 1 34
by Schwarz’ inequality
And therefore latently an “observable,” though originally intended to serve quite a different theoretical function; ρ is associated with the state of the ensemble, not with any device with which we may intend to probe the ensemble. One can, however, readily imagine a quantum “theory of measurement with devices of imperfect resolution” in which ρ -line constructs are associated with devices rather than states. 35 And will be the case if the ensemble came into being by action of a measurement device. 36 Such an assumption greatly simplifies certain arguments, but permits one to establish only weak instances of the general propositions in question.
33
Side conditions to which Wigner & Moyal functions are subject
But p2 + q 2 = (p + q)2 − 2pq = 1 − 2pq, so if we write (ψ|φ)(φ|ψ) ≡ cos2 θ we have trace ρ2 = 1 − 2pq sin2 θ ≤ 1 ↓ p = 1 & q = 0: the ensemble is pure; else p = 0 & q = 1: the ensemble is again pure; else = 1 if & only if sin θ = 0 : |ψ) ∼ |φ) so the ensemble is again pure Evidently "
pure mixed
ρ refers to a
#
" ensemble according as
trace ρ2 = 1 trace ρ2 < 1
# (69)
It follows that in the pure case ρ is projective; one has ρ2 = ρ ⇐⇒ trace ρ2 = 1 in the pure case
(70)
but to write ρ2 < ρ in the mixed case is to write (some frequently encountered) mathematical nonsense. The conclusions reached above hold generally (i.e., when the ensemble contains more than two states) but I will not linger to write out the demonstrations. Instead I look (because the topic is so seldom treated) to the spectral properties of ρ: Notice first that every state |ρ) which stands ⊥ to the space spanned by |ψ) and |φ) is killed by ρ —is, in other words, an eigenstate with zero eigenvalue: ρ |ρ) = 0
if |ρ) ⊥ both |ψ) and |φ)
The problem before us is, therefore, actually only 2-dimensional. Relative to some orthonormal basis {|1), |2)} in the 2-space spanned by |ψ) and |φ) we write $ $
ψ1 ψ2 φ1 φ2
% :
coordinate representation of |ψ)
:
coordinate representation of |φ)
%
In that language the associated projection operators Pψ and Pφ acquire the matrix representations $ Pψ ≡
ψ1 ψ1∗ ψ2 ψ1∗
ψ1 ψ2∗ ψ2 ψ2∗
%
$ and Pφ ≡
φ1 φ∗1 φ2 φ∗1
giving ρ −→ R = p Pψ + q Pφ . Looking now to det(R − ρ I) = ρ2 − ρ · trace R + det R
φ1 φ∗2 φ2 φ∗2
%
34
Status of Ehrenfest’s Theorem
we have trace R = p + q = 1 and, by quick calculation, det R = pq ψ1 ψ1∗ φ2 φ∗2 + ψ2 ψ2∗ φ1 φ∗1 − ψ1 φ∗1 φ2 ψ2∗ − ψ2 φ∗2 φ1 ψ1∗ = pq (ψ|ψ)(φ|φ) − (ψ|φ)(φ|ψ) = pq sin2 θ giving det(R − ρ I) = ρ2 − ρ + pq sin2 θ. The eigenvalues of R can therefore be described # ρ1 = 12 1 ± 1 − 4pq sin2 θ (71.1) ρ2 = 12 (p + q) ± (p + q)2 − 4pq sin2 θ (71.2) 1 2 2 = 2 (p + q) ± (p − q) + 4pq cos θ (71.3) Evidently each eigenvalue is real and non-negative sum of eigenvalues = p + q = 1
(72.1) (72.2)
I distinguish now several cases: • If pq sin2 θ = 0 because p (else q) vanishes37 —which is to say: if R is projective, and the ensemble therefore pure—then (71.1) gives ρ1 = 1
and ρ2 = 0
which conforms nicely to the general proposition that if P is projective ( P2 = P ) then det( P − λI ) = (1 − λ)dimension of image space · (0 − λ)dimension of its annihilated complement In the case (p = 1 & q = 0) we obtain descriptions of the associated eigenvectors $ R
ψ1 ψ2
%
$ =1·
ψ1 ψ2
%
$ and R
+ψ2∗ −ψ1∗
%
$ =0·
+ψ2∗ −ψ1∗
%
which are transparently orthonormal. Trivial adjustments yield statements appropriate to the complementary case (p = 0 & q = 1). • If pq = 0 but |ψ) ⊥ |φ) then (71.3) gives ρ1 = p 37
and ρ2 = q
I dismiss as physically uninteresting the possibility sin θ = 0, since it has been seen to lead to phony mixtures |φ) = ei(arbitrary phase) |ψ).
Side conditions to which Wigner & Moyal functions are subject
35
And it is under such circumstances evident that $ % $ % $ % $ % ψ1 ψ1 φ1 φ1 R =p· and R =q· ψ2 ψ2 φ2 φ2 • In the general case (pq = 0 & (ψ|φ) = 0) it becomes excessively tedious (even in the 2-dimensional case) to write out explicit descriptions of the eigenvectors. We are assured, however, that they exist, and are orthonormal, and that in terms of them the density matrix acquires a spectral representation of the form ρ = |ρ1 )ρ1 (ρ1 | + |ρ2 )ρ2 (ρ2 | +
∞
|ρk )0(ρk |
(73)
k=3
where |ρk ) is some/any basis in that portion of state space which annihilated by ρ, the space of states absent from the mixture to which ρ refers. But (73) permits/invites reconceptualization of the mixture: we imagine ourselves to have mixed states |ψ) and |φ) with probabilities p and q, but according to (73) we might equally well38 —in the sense that we would have obtained identical physical results if we had—mixed states |ρ1 ) and |ρ2 ) with probabilities ρ1 and ρ2 . Equation (73) describes an “equivalent mixture” which was “present like a spectre”39 in the original mixture, and which I will call the “ghost.” It is from the ghost that we acquire access to the “arguments from orthonormality” which 36 are standard to the literature, but which ! at the beginning of this discussion I was at pains to disallow; thus, taking to range over the ghost states present in the mixture, ρ= ↓ 2
ρ =
|ρk )ρk (ρk |
=⇒
trace ρ =
|ρk )ρ2k (ρk |
=⇒
trace ρ2 =
ρk = 1
(74.1)
ρ2k ≤ 1
(74.2)
with equality if an only if the mixture is in fact pure. Consistently with the latter claim: working from (71.1) we find
trace ρ2 = ρ21 + ρ22 = 1 − pq sin2 θ ≤ 1 which is precisely the result from which we extracted (69). 38
At least if this conjecture stands: Statements of the form (72) pertain generally—no matter how many states we mix, in what proportions, and no matter what may be the inner-product relationships among them. The point at issue is of an entirely mathematical—not a physical—nature, and will clearly require methods more powerful than the elementary methods that served me in the 2-dimensional case. 39 Recall that the word “spectrum” derives historically from Newton’s claim that colored light is “present like a spectre” in the mixture we call white light.
36
Status of Ehrenfest’s Theorem
We confront now this uncommon question: What mixtures are equivalent in the sense that—and physically indistinguishable because—they share the same ghost? We will suppose the states present in the mixture to span an n-dimensional space; in its ghostly representation the density matrix then reads ρ=
n
|ρk )ρk (ρk |
k=1
where by assumption none of the eigenvalues ρk vanishes. They are the roots, therefore, of a polynomial of the form n &
(ρ − ρk ) = ρn + c1 ρn−1 + · · · + cn−1 ρ + cn = 0
k=1
with c1 = −trace ρ = −1 and cn = (−)n det ρ = 0 (else 0 would join the set of eigenvalues, and the dimension of the mixture would be reduced). Now let {|ψ1 ), |ψ2 ), . . . , |ψn )} refer to any set of normalized states that span the mixture, and form the complex numbers zjk ≡ (ψj |ψk ) : j = k, which are n(n − 1) in number. Form n ρ˜ ≡ |ψk )pk (ψk | k=1
where the non-negative real numbers pk are subject to the constraint Construct the associated characteristic polynomial
!
pk = 1.
det (ρ 1 − ρ˜) = ρn − ρn−1 + c˜2 ρn−2 + · · · + c˜n−1 ρ + c˜n Specifically, we have where c˜2 = c˜2 (independent pi ’s and zjk ’s) .. . c˜n = c˜n (independent pi ’s and zjk ’s) To achieve equivalence in the strong sense ρ˜ = ρ it is necessary that ρ˜ and ρ have identical spectra; we are led therefore to these n − 1 conditions on a total of (n − 1) + n(n − 1) = n2 − 1 variables: c˜2 (independent pi ’s and zjk ’s) = c2 c˜3 (independent pi ’s and zjk ’s) = c3 .. . c˜n (independent pi ’s and zjk ’s) = cn
In the case n = 2 these become a single condition on three variables: p,
z12 ≡ (ψ1 |ψ2 )
& z21 ≡ (ψ2 |ψ1 )
(75)
Side conditions to which Wigner & Moyal functions are subject
37
Specifically (see again the calculation that led to (71)), we have p(1 − p) (ρ1 + ρ2 ) − z12 z21 = ρ1 ρ2 giving
√ ρ 1 ρ2 1 ± 1 − 4K with K ≡ and (ρ1 + ρ2 ) = 1 (ρ1 + ρ2 ) − z12 z21 '
= 12 (ρ1 + ρ2 ) ± (ρ1 + ρ2 ) − 4ρ1 ρ2 /(1 − z12 z21 )
p=
1 2
If in particular the selected states |ψ1 ) and |ψ2 ) happen to be orthogonal (if, in ∗ other words, z21 ≡ z12 = 0) then we recover p = ρ1
and q ≡ 1 − p = ρ2
and have achieved “weak equivalence” ρ˜ = |ψ1 )ρ1 (ψ1 | + |ψ2 )ρ2 (ψ2 |
∼
ρ = |ρ1 )ρ1 (ρ1 | + |ρ2 )ρ2 (ρ2 |